- 98-19 R. Paul
- A KAM Theorem for Some Degenerate Hamiltonian Systems
(85K, Latex 2e)
Jan 15, 98
(auto. generated ps),
of related papers
Abstract. We prove the existence of a large measure of invariant tori for
a class of perturbed integrable systems in which the unperturbed
Hamiltonian is degenerate (that is, its Hessian matrix does not
have full rank). This class consists of
systems for which--along with certain other restrictions--the unperturbed
Hamiltonian added to the average of the perturbation is nondegenerate.
Our result is similar to a theorem of Arnold, whose proof
is only sketched. In addition, we obtain explicit estimates on the measure
of phase space not foliated by invariant tori.
We apply our result to a system designed by Weinberg to test the
current theory of quantum mechanics. The system
is a perturbed simple harmonic oscillator, to which KAM
theorems with nondegeneracy conditions only on the unperturbed
Hamiltonian do not apply.